Graphics Reference
In-Depth Information
which interpolates between a series of points using repeated linear interpola-
tion.
9.6 B-Splines
B-splines
, like Bezier curves, use polynomials to generate a curve segment.
But, unlike Bezier curves, B-splines employ a series of control points that
determine the curve's local geometry. This feature ensures that only a small
portion of the curve is changed when a control point is moved.
There are two types of B-splines:
rational
and
non-rational
splines, which
divide into two further categories:
uniform
and
non-uniform
. Rational
B-splines are formed from the ratio of two polynomials such as
x
(
t
)=
X
(
t
)
W
(
t
)
,y
(
t
)=
Y
(
t
)
W
(
t
)
,z
(
t
)=
Z
(
t
)
W
(
t
)
,
Although this appears to introduce an unnecessary complication, the division
by a second polynomial brings certain advantages:
•
They describe perfect circles, ellipses, parabolas and hyperbolas, whereas
non-rational curves can only approximate these curves.
•
They are invariant of their control points when subjected to rotation,
scaling, translation and perspective transformations, whereas non-rational
curves lose this geometric integrity.
•
They allow weights to be used at the control points to push and pull the
curve.
An explanation of uniform and non-uniform types is best left until you
understand the idea of splines. So, without knowing the meaning of uniform,
let's begin with uniform B-splines.
9.6.1 Uniform B-Splines
A B-spline is constructed from a string of curve segments whose geometry
is determined by a group of local control points. These curves are known
as
piecewise polynomials
. A curve segment does not have to pass through a
control point, although this may be desirable at the two end-points.
Cubic B-splines
are very common, as they provide a geometry that is one
step away from simple quadratics, and possess continuity characteristics that
make the joins between the segments invisible. In order to understand their
construction, consider the scenario in Figure 9.14. Here we see a group of
(
m
+ 1) control points
P
0
,
P
1
,
P
2
,...,
P
m
which determine the shape of a
cubic curve constructed from a series of curve segments
S
0
,
S
1
,
S
2
,...,
S
m−
3
.
As the curve is cubic, curve segment
S
i
is influenced by
P
i
,
P
i
+1
,
P
i
+2
,
P
i
+3
,
and curve segment
S
i
+1
is influenced by
P
i
+1
,
P
i
+2
,
P
i
+3
,
P
i
+4
.Thereare
(
m
+ 1) control points, so there are (
m
−
2) curve segments.
